HAL Id: hal-03252856
https://hal.archives-ouvertes.fr/hal-03252856
Preprint submitted on 8 Jun 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
On a Set-Valued Young Integral with Applications to
Differential Inclusions
Laure Coutin, Nicolas Marie, Paul Raynaud de Fitte
To cite this version:
Laure Coutin, Nicolas Marie, Paul Raynaud de Fitte. On a Set-Valued Young Integral with
Applica-tions to Differential Inclusions. 2021. �hal-03252856�
On a set-valued Young integral with applications to
differential inclusions
Laure Coutina, Nicolas Marieb, Paul raynaud de Fittec
aIMT, Universit´e Paul Sabatier, Toulouse, France bMODAL’X, Universit´e Paris Nanterre, Nanterre, France
cLMRS, Universit´e de Rouen Normandie, Rouen, France
Abstract
We present a new Aumann-like integral for a H¨older multifunction with respect to a H¨older signal, based on the Young integral of a particular set of H¨older selections. This restricted Aumann integral has continuity properties that allow for numerical approximation as well as an existence theorem for an abstract stochastic differential inclusion. This is applied to concrete examples of first order and second order stochastic differential inclusions directed by fractional Brownian motion.
Keywords: set-valued integral, Aumann integral, Young integral
Contents
1 Introduction 2
2 Preliminaries 3
2.1 Notations and basic definitions . . . 3 2.2 Steiner point and generalized Steiner selections . . . 4 2.3 Young integral for single valued functions . . . 6
3 Set-valued Young integral 7
3.1 Special selections . . . 7 3.2 Aumann-Young integral . . . 10
4 Existence of fixed-points for functionals of Aumann-Young’s integral and applications to differential inclusions 17 4.1 Fixed point theorem . . . 18 4.2 Applications to differential inclusions . . . 21
Email addresses: [email protected] (Laure Coutin),
1. Introduction
Consider d, e ∈ N∗, a β-H¨older continuous signal w : [0, T ] → Rd with β ∈ (0, 1), and an α-H¨older continuous multifunction F defined on [0, T ] with convex compact values in the set Me,d(R) of linear mappings from Rd
to Re, where α ∈ (0, 1) and α + β > 1. The purpose of this paper is to define a set-valued Young integral of F with respect to w, of the form
Z T 0 F (s) dw(s) = ( Z T 0 f (s) dw(s) ; f ∈ S(F ) ) (1)
in a nontrivial way, but with a small enough set of selections S(F ), so as to get algebraic and topological properties (convexity, boundedness, compactness, continuity, etc.) allowing to give a sense and establish the existence of solutions to several types of differential inclusions.
There have been many different approaches to set-valued integration with respect to a nonnegative σ-additive measure µ. The most popular one is due to Aumann [4], based on Lebesgue integrals of selections:
Z T 0 F (s) dµ(s) = ( Z T 0 f (s) dµ(s) ; f ∈ SL1(µ)(F ) ) (2)
where SL1(µ)(F ) denotes the set of all µ-integrable selections of F . In the
case of multifunctions with convex compact values, other approaches such as Hukuhara’s [15] or Debreu’s [10] are restricted to multifunctions with compact convex values and use the cone structure of the space of convex compact sets.
The concept of Aumann integral has been applied in several papers to in-tegration of multivalued stochastic processes using classical stochastic calculus and a definition of the form (1), where w is a Brownian motion, or more gen-erally a semimartingale, e.g., [18, 22, 23]. Despite this similarity, the setting of stochastic calculus is quite different from ours, since the stochastic integral needs a probability space to make sense. In this context, some variants have been developped, the main one by Jung and Kim [16], which is the decompos-able hull of an integral of the form (1), has been extensively studied by Polish mathematicians from Zielona G´ora [19, 20, 17, 24, 25], to cite but a few papers and a book.
In the case of a deterministic signal w with possibly infinite variation, Michta and Motyl [26, 27] are the only references so far defining a set-valued Young integral `a la Aumann of the form (1), for convex as well as nonconvex-valued multifunctions. In their approach, the set of selections S(F ) is large, namely, in the case of our setting, S(F ) is the set of all α-H¨older continuous selections of F . Of course this is a natural definition, and the authors obtain basic expected properties on the set-valued integral: nonemptyness, convexity and regularity of the integral with respect to S(F ) (not F ). In our approach, the set of selections is smaller:
The ”tuning parameter” r > 0 controls both the α-H¨older seminorm of F and of the considered selections. This allows to establish the compactness of the integral, to get the upper semicontinuity of the integral with respect to F , and then to establish the existence of solutions to some differential inclusions. Note that our integral converges to that of Michta and Motyl [26] when the tuning parameter r goes to +∞. However, our integral is always compact-valued, whereas that of [26] may be unbounded, see Example 3.11 below.
On differential inclusions driven by α-H¨older continuous signals, let us men-tion Bailleul et al. [6]. In this paper, the authors establish the existence of solutions to a differential inclusion using the approach of Aubin and Cellina [2]. Let us also cite Levakov and Vas’kovskii [21] who mix pathwise integration with respect to fractional Brownian motion with Itˆo’s integral with respect to standard Brownian motion, following Guerra and Nualart [1]. These works on differential inclusions implicitely use an Aumann type set-valued integral of the form (1).
As an application of our set-valued Young integral, we are able to define a stochastic set-valued integral with respect to the fractional Brownian motion (fBm) of Hurst index H > 1/2, and then to establish the existence of solutions to several types of stochastic differential inclusions driven by the fBm.
This paper is organized as follows. Section 2 recalls preliminary definitions and results on Steiner’s selections and on the Young integral for point-valued functions. Section 3 deals with the set-valued Young integral. Finally, Section 4 provides a fixed-point theorem for functionals of the set-valued Young integral which allows, in particular, to get the existence of solutions to several types of differential inclusions.
2. Preliminaries
2.1. Notations and basic definitions Let d > 1 be an integer.
1. The set of nonempty closed subsets of Rd is denoted by Pf(Rd). The semi-Hausdorff distance on Pf(Rd) is denoted by dH: for all (A, B) ∈ Pf(Rd)2
dH(A, B) = max sup a∈A d(a, B) ; sup b∈B d(b, A) = sup x∈Rd |d(x, A) − d(x, B)| (3)
(see, e.g., [7]). For every A ∈ Pf(Rd), we denote kAk
dH := dH(A, {0Rd}) = sup{kak ; a ∈ A}.
2. Let Pck(Rd
) be the space of nonempty, convex and compact subsets of Rd. For any C ∈ Pck(Rd), the support function of C is the map
δ∗(., C) :
Rd → R
We have, for all A, B ∈ Pck(Rd), dH(A, B) = sup `∈Rd,k`k=1 |δ∗(`, A) − δ∗(`, B)| . For any C ∈ Pck(Rd ) and ` ∈ Rd, consider Y (`, C) := {c ∈ C : h`, ci = δ∗(`, C)} .
If Y (`, C) contains exactly one element, it is denoted by y(`, C) and called exposed point of C with exposing direction `. The set of all exposing direc-tions for a given point of C is denoted by TC. For all A, B ∈ Pck(Rd), the Demyanov distance between A and B is defined by
dD(A, B) := sup{ky(`, A) − y(`, B)k ; ` ∈ TA∩ TB}.
Note that dD(A, B) > dH(A, B) for all A, B ∈ Pck(Rd), see, e.g., [29]. On Rd, which we identify with the subset of singletons of Pck(Rd), both distances dD and dHcoincide with the distance induced by k.k.
3. For any α ∈]0, 1[, Cα-H¨ol([0, T ]; Pck(Rd)) (respectively Cα-H¨D ol([0, T ]; Pck(Rd))) is the space of α-H¨older continuous maps from [0, T ] into Pck(Rd), where Pck(Rd) is endowed with the Hausdorff distance (respectively, the Demyanov distance). In the sequel, Cα-H¨ol([0, T ]; P
ck(Rd)) is equipped with the α-H¨older norm Nα,T(.) := k.k∞,T + k.kα,T, where, for F ∈ Cα-H¨ol([0, T ]; Pck(Rd)),
kF kα,T = sup dH(F (s), F (t))
|t − s|α ; s, t ∈ [0, T ] and s < t
,
and (with inconsistent but convenient notations)
kF k∞,T = sup t∈[0,T ]
kF (t)kd
H.
We denote by Bα,Pck(f, δ) the closed ball of center f and radius δ in the
space Cα-H¨ol([0, T ]; P
ck(Rd)) . Similarly, the space Cα-H¨ol
Dem ([0, T ]; Pck(Rd)) is equipped with the α-H¨older norm Nα,T ,Dem(.) := k.k∞,T+ k.kα,T ,Dem, where dHis replaced by dDin the above definition.
4. For any s, t ∈ [0, T ] such that t > s, D[s,t] is the set of all dissections of [s, t]. 2.2. Steiner point and generalized Steiner selections
Let
M := {µ probability measure on BRd(0, 1)
Definition 2.1. The Steiner point of C ∈ Pck(Rd) is defined by St(C) := 1 vd Z TC∩BRd(0,1) y(x, C) dx with vd= π d/2 Γ(1 + d/2).
It is well known that St(C) ∈ C for any C ∈ Pck(Rd) (see the more general Proposition 2.5 below). Furthermore, we have:
Proposition 2.2 (Lipschitz property). The map St : Pck(Rd) 7→ Rd is kd-Lipschitz for the Hausdorff distance dH, where the sharp kd-Lipschitz coefficient kd satisfies r 2d π < kd< r 2(d + 1) π . (4)
See [31, 30] for the exact calculation of kd. The estimation (4) can be found in [30].
The Steiner point can be generalized by replacing the probability measure dx/vdin Definition 2.1 by any element of M. This is particularly interesting in connection with the Demyanov distance.
Definition 2.3 (generalized Steiner selection). The Generalized Steiner point of C ∈ Pck(Rd), for a measure µ ∈ M, is defined by
Stµ(C) := Z
B
Rd(0,1)
St(Y (x, C))µ(dx).
The generalized Steiner selection of a multifunction F : [0, T ] → Pck(Rd) with respect to a measure µ ∈ M, is the map t ∈ [0, T ] 7→ Stµ(F (t)).
Proposition 2.4. For all C, C1, C2 ∈ Pck(Rd), µ1, µ2, µ ∈ M, κ ∈ [0, 1], λ, ν > 0, we have
Stκµ1+(1−κ)µ2(C) = κStµ1(C) + (1 − κ)Stµ2(C),
Stµ(λC1+ νC2) = λStµ(C1) + νStµ(C2). See Baier and Farkhi [5, Lemma 4.1] for a proof.
Theorem 2.5. (Castaing representation) For every measurable multifunction F : [0, T ] → Pck(Rd), there exists a sequence (µ
n)n∈N of elements of M such that, for every t ∈ [0, T ],
F (t) = [ n∈N
{Stµn(F (t))}.
See Dentcheva [11, Theorem 3.4] for a proof.
It is proved in [11] that, for any µ ∈ M, the map Stµ : Pck(Rd
) 7→ Rd is Lipschitz for the Hausdorff distance dH. However there is no uniform bound on the Lipschitz coefficient with respect to µ. But the Demyanov distance dD can be expressed using generalized Steiner points:
Proposition 2.6 (Demyanov distance and generalized Steiner points). For ev-ery C1, C2∈ Pck(Rd),
dD(C1, C2) = sup µ∈M
kStµ(C2) − Stµ(C2)k.
See Baier and Farkhi [5, Corollary 4.8] for a proof.
2.3. Young integral for single valued functions
This subsection deals with the definition and some basic properties of Young integral which allow to integrate a map f ∈ Cα-H¨ol([0, T ]; Me,d(R)) with respect to w when α ∈]0, 1[ and α + β > 1. Here, Me,d(R) denotes the space of real matrices with e rows and d columns.
Theorem 2.7 (Young integral). Consider α, β ∈]0, 1] such that α + β > 1, and two maps
f ∈ Cα-H¨ol([0, T ]; Me,d(R)) and w ∈ Cβ-H¨ol([0, T ]; Rd). For every n ∈ N∗ and Dn = (tn
1, . . . , tnmn) ∈ D[0,T ] such that |Dn| → 0 (where
|Dn| = max16k6mn−1(tnk+1− tnk)) the limit
lim n→∞ mn−1 X k=1 f (tnk)(w(tnk+1) − w(tnk))
exists and does not depend on the dissection Dn.
Definition 2.8 (Young integral). The limit in Theorem 2.7 is denoted by
Z T 0
f (s) dw(s)
and called the Young integral of f with respect to w on [0, T ].
The following theorem provides a bound on Young’s integral which is crucial in the sequel.
Theorem 2.9 (Young-Love estimate). Consider α, β ∈]0, 1[ such that α+β > 1, and two maps
f ∈ Cα-H¨ol([0, T ]; Me,d(R)) and w ∈ Cβ-H¨ol([0, T ]; Rd).
There exists a constant cα,β > 1, depending only on α and β, such that, for all s, t ∈ [0, T ] such that s < t, Z t s f (u) dw(u) − f (s)(w(t) − w(s)) 6 c α,βkwkβ,Tkf kα,T|t − s|α+β.
Therefore, for all s, t ∈ [0, T ], Z t s f (u) dw(u) 6 c α,βkwkβ,T(kf kα,TTα+ kf k∞,T)|t − s|β, and, in particular, Z . 0 f (s) dw(s) β,T 6 C α,βkwkβ,TNα,T(f ) with Cα,β,T = cα,β(Tα∨ 1).
See Friz and Victoir [12, Theorem 6.8] for a proof.
3. Set-valued Young integral
This section deals with an Aumann-like integral based on a special subset of selections. We assume the following hypothesis:
(A) F ∈ Cα-H¨ol([0, T ]; Pck(Me,d(R))), and w ∈ Cβ-H¨ol([0, T ]; Rd), with α, β ∈ ]0, 1], α + β > 1.
We shall sometimes consider the subcase obtained by adding the following stronger assumption on F :
(B) F ∈ Cα-H¨ol
Dem ([0, T ]; Pck(Me,d(R))).
To show that (B) is not contained in (A), we can consider the case when F (t) is the segment of R2 with one end at (0; 0) and the other end at (sin t, cos t) (see [29, Example 3.3]). Then dD(F (t), F (s)) > 1 for s 6= t, thus Nα,T ,Dem(F ) = +∞ for all α, whereas F is Lipschitz for dH.
3.1. Special selections
We now define an appropriate set of selections of F , that will be used to define a set-valued Young integral of F with respect to w.
Let us choose our ”tuning parameter” r such that
r > kedkF kα,T, (5) where ked is the constant defined in Proposition 2.2. In the case when (B) is satisfied, we can alternatively take
r > kF kα,T ,Dem. (6) Note that Condition (6) can be less restrictive than (5), when (B) is satisfied. For example, if F (t) has the form f (t)+C, where f is single-valued and C ∈ Pck(Rd) is constant, we have kF kα,T ,Dem= kF kα,T 6 kedkF kα,T.
Notation 3.1. In this section, we denote rmin= min kedkF kα,T, kF kα,T ,Dem . (7)
Notation 3.2. We denote by S0(F ) the set of all measurable selections of F , Sα,r(F ) := {f ∈ S0(F ) ; kf kα,T 6 r} and
SSt(F ) := {Stµ(F (.)) ; µ ∈ M} ⊂ S0(F ).
Remark 3.3 (Dependence on T of Sα,r(F )). Let t ∈ [0, T [. If f is an α-H¨older selection of F on [0, t], with kf kα,t6 r, there does not necesarily exists a selection f ∈ Sα,r(F ) wich extends f on [0, T ] and such that f α,T 6 r. So, Sα,r(F ) depends on T , but we chose to keep the relatively light notation Sα,r(F ) without stressing this fact.
Proposition 3.4. For every r > rmin, the set Sα,r(F ) is nonempty and convex. Proof. The convexity of Sα,r(F ) stems from the convexity of the norm Nα,T.
If (5) is satisfied, Sα,r(F ) is nonempty because it contains St(F ) by Propo-sition 2.2. Indeed, we have, for s, t ∈ [0, T ],
kSt(F (t)) − St(F (s))k 6 keddH(F (t), F (s)) 6 kedkF kα,T|t − s|α, thus kSt(F )kα,T 6 r.
Similarly, if (B) and (6) are satisfied, we have St(F ) ⊂ Sα,r(F ) by Proposi-tion 2.6.
Remark 3.5 (Basic properties of the selections sets).
1. The set SSt(F ) is a nonempty and convex subset of Cα-H¨ol([0, T ]; Me,d(R)) such that, for every f ∈ S(F ), Nα,T(f ) 6 Nα,T ,Dem(F ). Moreover, there exists a sequence (fn)n∈N of elements of SSt(F ) such that, for every t ∈ [0, T ],
F (t) = [ n∈N
{fn(t)}.
Indeed, this follows from Propositions 2.6, 2.5 and 2.4.
2. Consequently, if (B) and (6) are satisfied, we have SSt(F ) ⊂ Sα,r(F ) and there exists a sequence (fn)n∈Nof elements of Sα,r(F ) such that, for every t ∈ [0, T ],
F (t) = [ n∈N
{fn(t)}. (8)
Let us establish some topological properties of Sα,r(F ) in L2([0, T ]; Me,d (R)) and in Cα-H¨ol([0, T ]; Me,d
Proposition 3.6. For every r > rmin, the set Sα,r(F ) is a bounded and closed subset of Cα-H¨ol([0, T ]; Me,d(R)). Moreover, for every sequence (fn)n∈N of ele-ments of Sα,r(F ), there exists a subsequence (fnk)k∈N of (fn)n∈N such that, for every ε ∈]0, α], (fnk)k∈N converges in C(α−ε)-H¨ol([0, T ]; Me,d(R)) to an element of Sα,r(F ).
Proof. Let us prove each property of the set Sα,r(F ) stated in Proposition 3.6. Consider f ∈ Sα,r(F ). Then, Nα,T(f ) 6 kF k∞,T + r. Since r does not depend on f , Sα,r(F ) is a bounded subset of Cα-H¨ol([0, T ]; Me,d
(R)).
Consider a sequence (fn)n∈N of elements of Sα,r(F ) which converges in Cα-H¨ol([0, T ]; Me,d(R)) (equipped with Nα,T). Its limit is denoted by f . In particular,
lim
n→∞kfn(t) − f (t)k = 0 ; ∀t ∈ [0, T ]. (9) On the one hand,
Nα,T(f ) 6 sup n∈N
Nα,T(fn) 6 kF k∞,T+ r.
On the other hand, for every t ∈ [0, T ], since F (t) is a closed subset of Me,d(R), and since fn(t) ∈ F (t) for every n ∈ N, f (t) ∈ F (t) by the convergence result (9). So, f ∈ Sα,r(F ), and then Sα,r(F ) is a closed subset of Cα-H¨ol([0, T ]; Me,d
(R)). Let (fn)n∈N be a sequence of elements of Sα,r(F ). By the definition of Sα,r(F ), sup n∈N kfnk∞,T 6 kF k∞,T and sup n∈N kfnkα,T 6 r.
Therefore, by Friz and Victoir [12, Proposition 5.28], there exists a subse-quence (fnk)k∈N of (fn)n∈Nsuch that, for every ε ∈]0, α], (fnk)k∈Nconverges in C(α−ε)-H¨ol([0, T ]; Me,d(R)) to an element of Sα,r(F ).
Proposition 3.7. Sα,r(F ) is a bounded, closed and sequentially compact subset of L2([0, T ]; Me,d
(R)).
Proof. Let us prove each property of the set Sα,r(F ) stated in Proposition 3.7. Consider f ∈ S0(F ). For every t ∈ [0, T ],
kf (t)k 6 sup{kxk ; x ∈ F (t)} = kF (t)kdH 6 kF k∞,T.
Then,
kf kL2([0,T ])6 T1/2kf k∞,T 6 T1/2kF k∞,T.
Therefore, Sα,r(F ) ⊂ S0(F ) is bounded in L2([0, T ]; Rd).
Consider a sequence (fn)n∈N of elements of Sα,r(F ) converging in L2([0, T ]; Me,d(R)). Its limit is denoted by f . Then, there exists a subsequence (fnk)k∈N
of (fn)n∈Nsuch that lim
k→∞kfnk(t) − f (t)k = 0 ; ∀t ∈ [0, T ]. (10) On the one hand, for every t ∈ [0, T ], since F (t) is a closed subset of Me,d(R), and since fnk(t) ∈ F (t) for every k ∈ N, we have f (t) ∈ F (t) by the convergence
result (10). On the other hand, by Proposition 3.6, we can choose (fnk)k∈Nsuch that, for every ε ∈]0, α], (fnk)k∈N converges in C(α−ε)-H¨ol([0, T ]; Me,d(R)), and then f ∈ Sα,r(F ). Thus, Sα,r(F ) is a closed subset of L2([0, T ]; Me,d(R)).
Let (fn)n∈Nbe a sequence of elements of Sα,r(F ). According to Proposition 9, there exists a subsequence (fnk)k∈Nof (fn)n∈N such that, for every ε ∈]0, α], (fnk)k∈Nconverges in C
(α−ε)-H¨ol([0, T ]; M
e,d(R)) to an element of Sα,r(F ). Note that
sup k
kfnkk∞,T 6 kF k∞,T < ∞. (11)
Then, the sequence (fnk)k is uniformly integrable with respect to the Lebesgue measure on [0, T ] and (fnk)k converges to f in L1([0, T ]; Me,d
(R)). Using esti-mation (11), the convergence holds in L2([0, T ]; Me,d
(R)). Thus, Sα,r(F ) is a sequentially compact subset of L2([0, T ]; Me,d
(R)).
3.2. Aumann-Young integral Now, consider w ∈ Cβ-H¨ol
([0, T ]; Rd) and let us define a set-valued Young integral with respect to w, using special sets of selections.
Definition 3.8 (Aumann-Young integral). The Aumann-Young integral of F with respect to w and parameters α and r is defined by
(Aα,r) Z T 0 F (t) dw(t) = JT ,α,r(F, w) := ( Z T 0 f (t) dw(t) ; f ∈ Sα,r(F ) ) .
Remark 3.9. Michta and Motyl define a larger Aumann-Young integral in [26]. For convex-valued F , their integral is
(Aα,+∞) Z T 0,+∞ F (t) dw(t) = JT ,α,+∞(F, w) := ( Z T 0 f (t) dw(t) ; f ∈ S0(F ) ∩ Cα-H¨ol([0, T ]; Me,d(R)) ) = [ r>0 JT ,α,r(F, w).
See Example 3.11 below for a comparison with (Aα,r)R0,+∞T F (t) dw(t) when r < ∞.
The following proposition provides some basic properties of the set-valued Aumann-Young integral.
Proposition 3.10. For every r > rmin, the Aumann-Young integral of F with respect to w and parameters α and r is a nonempty, bounded, closed and convex subset of Re.
Proof. Let us prove each property of the Aumann-Young integral of F with respect to w stated in Proposition 3.10.
Since Sα,r(F ) is nonempty (resp. convex), JT ,α,r(F, w) is nonempty (resp. con-vex).
By Theorem 2.9, for every f ∈ Sα,r(F ), Z T 0 f (t) dw(t) 6cα,βTβkwkβ,T(kf kα,TTα+ kf k∞,T 6cα,βTβ(Tα∨ 1)kwkβ,T(r + kF k∞,T).
Then, the Aumann-Young integral of F with respect to w is a bounded subset of Re.
Consider a converging sequence (jn)n∈Nof elements of JT ,α,r(F, w). Its limit is denoted by j. By the definition of JT ,α,r(F, w), for every n ∈ N, there exists fn∈ Sα,r(F ) such that
jn= Z T
0
fn(t) dw(t).
By Proposition 3.6, there exists a subsequence (fnk)k∈N of (fn)n∈N such that, for any ε ∈]0, α], (fnk)k∈Nconverges in C(α−ε)-H¨ol([0, T ]; Me,d
(R)) to an element f of Sα,r(F ). So, by Theorem 2.9, for ε < α + β − 1, for every k ∈ N,
jnk− Z T 0 f (t) dw(t) 6cα,βkwkβ,TTβ(kfnk− f kα−ε,TT α−ε+ kf nk− f k∞,T) 6cα,βkwkβ,TTβ(Tα−ε∨ 1)Nα−ε,T(fnk− f ) −−−−→ k→∞ 0. Therefore, j = Z T 0 f (t) dw(t),
and then JT ,α,r(F, w) is a closed subset of Re.
Example 3.11 (Comparison with Michta and Motyl’s Young integral). In this example, Michta and Motyl’s Young integral [26] is equal to R, whereas, by Proposition 3.10, JT ,α,r(F, w) is a compact interval. Assume that β 6 1
2, and let
w(t) = t2βcos(π/t) and F (t) = [−1, 1], (t ∈ [0, 1]). We have Nα,T(F ) = Nα,T ,Dem(F ) = 1 and, for 0 6 s < t 6 1,
|w(t) − w(s)| 6 t2β− s2β cos(π/t) + s2β(cos(π/t) − cos(π/s)) 6 t2β− s2β
6 t2β− s2β + 21−β πβs2β 1 sβ − 1 tβ 62 tβ− sβ + 21−βπβ tβ− sβ 6 (t − s)β 2 + 21−βπβ , thus kwkβ,16 2 + 21−βπβ.
Let us define a sequence (fn)n>1of selections of F by
fn(t) =
sin(π/t) if n1 6 t 6 1 0 if 0 6 t 6n1. Clearly, kfnkα,1→ +∞ when n → ∞. Furthermore,
Z 1 0 fn(t) dw(t) = Z 1 1/n sin(π/t) d t2βcos(π/t) = Z 1 1/n
sin(π/t)2βt2β−1cos(π/t) − πt2β−2sin(π/t)dt
= Z n
1
sin(πu)2βu1−2βcos(πu) − πu2−2βsin(πu) 1 u2du 6 2β Z n 1 u−1−2β− 2 n−1 X k=1 (k + 1)−2β Z k+1 k sin2(πu) du → −∞ when n → ∞.
By convexity of the Aumann integral and symmetry of F , this shows that the integral of F with respect to w in the sense of Michta and Motyl [26] is the whole line R.
Proposition 3.12 (Lipschitz continuity result with respect to the driving sig-nal). For every r > rmin, the set-valued map
JT ,α,r(F, .) : Cβ-H¨ol ([0, T ]; Rd) −→ P ck(Re) w 7−→ JT ,α,r(F, w)
is Lipschitz continuous when Pck(Re) is endowed with the Hausdorff distance dH.
Proof. Consider w1, w2 ∈ Cβ-H¨ol
([0, T ]; Rd) and j1 ∈ JT ,α,r(F, w1). So, there exists f1∈ Sα,r(F ) such that
j1= Z T
0
f1(s) dw1(s),
d(j1, JT ,α,r(F, w2)) = inf f ∈Sα,r(F ) Z T 0 f (t) dw2(t) − Z T 0 f1(t) dw1(t) 6 Z T 0 f1(t)d(w2− w1)(t) + inf f ∈Sα,r(F ) Z T 0 (f − f1)(t) dw2(t) 6cα,βTβ(Tα∨ 1) Nα,T(f1)kw1− w2kβ,T + kw2kβ,T inf f ∈Sα,r(F ) Nα,T(f − f1) . Since f1∈ Sα,r(F ), Nα,T(f1
) 6 r + kF k∞,T and the second term in the right-hand side of the previous inequality is null. Then
d(j1, JT ,α,r(F, w2)) 6 cα,βTβ(Tα∨ 1)(r + kF k∞,T)kw 1− w2k
β,T and, by symmetry,
d(j2, JT ,α,r(F, w1)) 6 cα,βTβ(Tα∨ 1)(r + kF k∞,T)kw1− w2kβ,T for every j2∈ JT ,α,r(F, w2). Therefore,
dH(JT ,α,r(F, w1), JT ,α,r(F, w2)) = max ( sup j1∈J T ,α,r(F,w1) d(j1, JT ,α,r(F, w2)) ; sup j2∈J T ,α,r(F,w2) d(j2, JT ,α,r(F, w1)) ) 6 cα,βTβ(Tα∨ 1)(r + kF k∞,T)kw1− w2kβ,T.
The following proposition provides semicontinuity results for the set-valued Young’s integral. Let us first recall the topological superior and inferior limits in Kuratowski’s sense for a sequence of sets, see, e.g., [7]. If (An) is a sequence of closed subsets of a metric space M, let us denote
Li
n→∞An the set of limits of sequences (xn) such that xn ∈ An for every n,
Ls
n→∞An the set of limits of sequences (xn) such that xn∈ Amn for every n for some subsequence (Amn) of (An).
Clearly, Li
n→∞An⊂ Lsn→∞An. We say that (An) converges in Kuratowski’s sense to A ⊂ M if
Ls
n→∞An⊂ A ⊂ Lin→∞An.
Convergence in Kuratowski’s sense is weaker than convergence for the Hausdorff distance, however both convergences are equivalent if M is compact. Indeed, by, e.g., [8, Theorem 3.1 page 51], the set Pk(M) of compact subsets of M is compact for the topology of Hausdorff distance, thus this topology coincide with any weaker separated (T2) topology on Pk(M). But, if M is compact, the convergence in Kuratowski’s sense is associated with a separated topology (see, e.g., [7, Theorem 5.2.6]).
Proposition 3.13 (Semicontinuity with respect to F ). Let r > rmin.
1. Let (Fn)n∈N be a sequence of elements of Cα-H¨ol([0, T ]; Pck(Me,d(R))) such that
Ls
n→∞Fn(t) ⊂ F (t) ; ∀t ∈ [0, T ], and that
Fn∈ Bα,Pck(Me,d(R))(0, r + εn)
for every n ∈ N, for some sequence (εn)n∈N of elements of R+, converging to 0. Then
Ls
n→∞JT ,α,r+εn(Fn, w) ⊂ JT ,α,r(F, w). 2. Assume furthermore that
F (t) ⊂ Li
n→∞Fn(t) ; ∀t ∈ [0, T ]. Then
JT ,α,r(F, w) ⊂ Li
n→∞JT ,α,2r+n(Fn, w). Proof. 1. Let us prove that
J := Ls
n→∞JT ,α,r+εn(Fn, w) ⊂ JT ,α,r(F, w).
Consider j ∈ J . Then, there exists a sequence (fn)n∈N of elements of the space Cα-H¨ol([0, T ]; Me,d (R)) such that fn ∈ Sα,r+εn(Fn) ; ∀n ∈ N and j = lim k→∞ Z T 0 fnk(t) dw(t),
where (fnk)k∈N is a subsequence of (fn)n∈N. By the definition of Sα,r+εn(Fn),
n ∈ N, kfnk(t)k 6 sup n,s kFn(s)kdH 6 kF k∞,T+ sup n∈N εn< ∞ for every k ∈ N and t ∈ [0, T ], and
sup k∈N
kfnkkα,T 6 r + sup
n∈N εn.
Then, by Friz and Victoir [12, Proposition 5.28], there exists a subsequence (fmk)k∈N of (fnk)k∈N such that, for every ε ∈]0, α], such that ε < α + β − 1, (fmk)k∈Nconverges in C(α−ε)-H¨ol([0, T ]; Rd) to an element f of Cα-H¨ol([0, T ]; Rd). So, j = Z T 0 f (t) dw(t).
It remains to check that f ∈ Sα,r(F ). For any t ∈ [0, T ], since fmk(t) ∈ Fmk(t) for every k ∈ N, and since f is in particular the pointwise limit of (fmk)k∈N,
f (t) = lim sup k→∞
fmk(t) ∈ Ls
k→∞Fmk(t) = F (t). Moreover, by Friz and Victoir [12, Lemma 5.12],
kf kα,T 6 lim inf
k→∞ kfmkkα,T 6 r + limn→∞εn= r. Therefore, j ∈ JT ,α,r(F, w).
2. The supplementary hypothesis implies that (Fn(t)) converges in Kura-towski’s sense to F (t). Furthermore, since (Fn) is bounded in Cα-H¨ol([0, T ]; Pck(Me,d(R))), it is bounded for k.k∞,T, thus (Fn(t)) converges to F (t) for the Hausdorff distance.
Let j ∈ JT ,α,r(F, w), and let f ∈ Sα,r(F ) such that j =R0Tf (t) dw(t). Set fn(t) = πFn(t)(f (t)) for every t ∈ [0, T ] and each integer n, where πFn(t)denotes
the orthogonal projection on Fn(t). We have, for any t ∈ [0, T ], kf (t) − fn(t)k 6 dH(F (t), Fn(t)) → 0 when n → ∞,
thus (fn) converges uniformly to f . Furthermore, for any n and for s, t ∈ [0, T ], kfn(t) − fn(s)k 6 πFn(t)(f (t)) − πFn(t)(f (s)) +
πFn(t)(f (s)) − πFn(s)(f (s)) 6 kf (t) − f (s)k + dH(Fn(t), Fn(s)).
Indeed, it is well known that the projection operator πFn(t) is non expansive
(see, e.g., [14, page 118]), and the estimation of
πFn(t)(f (s)) − πFn(s)(f (s))
follows from (3). Since r > kF kα,T, we deduce
kfnkα,T 6 kf kα,T + kFnkα,T 6 2r + n. Thus jn := RT
0 fn(t) dw(t) ∈ JT ,α,2r+n(Fn, w). Furthermore, thanks to [12,
Proposition 6.12], we have limn→∞jn= j.
The preceding result can be improved when the multifunctions Fn are con-structed from F using some recipe which can also be applied to their selections. This can be useful for numerical approximations.
Proposition 3.14. (Time discretization of the multivalued integral) Let (Dn) be a sequence of dissections of [0, T ], say, Dn = (tn0, . . . , tnmn), 0 = t0< · · · < tnmn = T , and assume that |Dn| converges to 0, where |Dn| = max16i6mn−1(tni+1− tn
i) is the mesh of Dn. For each n, for each i ∈ {1, . . . , mn − 1} and for any t ∈ [tni, tni+1], set Fn(t) = t − tn i tn i+1− t n i F (tni) + t n i+1− t tn i+1− t n i F (tni+1). Then lim n→∞dH(JT ,r(Fn, w), JT ,r(F, w)) = 0.
Proof. By uniform continuity of F on [0, T ], the sequence (Fn) converges uni-formly to F for dH, and kFnkd
H 6 kF kdH for all n. Furthermore, we have
Nα,T(Fn) 6 Nα,T(F ) for all n, see [9].
From Part 1 of Proposition 3.13, we have that Ls JT ,r(Fn, w) ⊂ JT ,α,r(F, w). Now, let j ∈ JT ,α,r(F, w), and let f ∈ Sα,r(F ) such that j =RT
0 f (t) dw(t). Define fn ∈ Sα,r(Fn) by fn(t) = t − t n i tn i+1− tni f (tni) + tni+1− t tn i+1− tni f (tni+1).
for each i ∈ {1, . . . , mn − 1} and for every t ∈ [tn
i, tni+1]. Then (fn) con-verges uniformly to f , and we conclude as in the proof of Proposition 3.13 that JT ,α,r(F, w) ⊂ Li JT ,r(Fn, w), thus (JT ,r(Fn, w))nconverges to JT ,r(Fn, w) in Kuratowski’s sense.
Since the sequence (JT ,r(Fn, w))n is bounded for k.kd
H, it is relatively
com-pact for the Hausdorff distance, we deduce that it converges to JT ,r(Fn, w) for dH.
Let us conclude this section by investigating the indefinite Aumann-Young integral t 7→ (Aα,r) Z t 0 F (s) dw(s) = Jt,α,r(F, w) := ( Z T 0 f (s)1[0,t](s) dw(s) ; f ∈ Sα,r(F ) ) .
Remark 3.15 (Dependence on T of the indefinite integral). Since (Aα,r) Rt
0F (s) dw(s) is built using elements of Sα,r(F ), it follows from Remark 3.3 that our indefi-nite integral depends on T . More accurate but rather heavy notations could be (Aα,T ,r)
Rt
0F (s) dw(s) = Jt,α,T ,r(F, w).
Remark 3.16. The previous results on JT ,α,r(F, w) remain true for Jt,α,r(F, w), by the same arguments.
Proposition 3.17 (Continuity of the indefinite Aumann-Young integral). The set-valued map t ∈ [0, T ] 7→ Jt,α,r(F, w) is β-H¨older continuous with constant Cα,β,T(kF k∞,T+r)kwkβ,T when Pck(Re) is endowed with the Hausdorff distance dH, where Cα,β,T is the constant defined in Theorem 2.9.
Proof. Consider s, t ∈ [0, T ] with s < t, and jt∈ Js,α,r(F, w). So, there exists ft∈ Sα,r(F ) such that jt= Z t 0 ft(u) dw(u), and by Theorem 2.9, d(jt, Js,α,r(F, w)) = inf Z t ft(u) dw(u) − Z s f (u) dw(u)
6 Z t s ft(u) dw(u) + inf f ∈Sα,r(F ) Z s 0 (f − ft)(u) dw(u) 6cα,β(Tα∨ 1)kwkβ,T Nα,T(ft)|t − s|β+ Tβ inf f ∈Sα,r(F ) Nα,T(f − ft) .
Since fs ∈ Sα,r(F ), we have Nα,T(ft) 6 (kF k∞,T + r) and the second term in the right-hand side of the previous inequality is null. Then,
d(jt, Js,α,r(F, w)) 6 cα,β(Tα∨ 1)kwkβ,T(kF k∞,T + r)|t − s|β and, by symmetry,
d(jt, Js,α,r(F, w)) 6 cα,β(Tα∨ 1)kwkβ,T(kF k∞,T + r)|t − s|β for every jt∈ Jt,α,r(F, w). Therefore,
dH(Js,α,r(F, w), Jt,α,r(F, w)) = max ( sup js∈Js,α,r(F,w) d(js, Jt,α,r(F, w)) ; sup jt∈Jt,α,r(F,w) d(jt, Js,α,r(F, w)) ) 6 Cα,β,Tkwkβ,T(kF k∞,T + r)|t − s|β.
Corollary 3.18 (Upper bound for kJ.,α,r(F, w)kα,T and Nα,T(J.,α,r(F, w))). Assume that r > rmin, and let ρw(T, r) := Cα,β,T(kF k∞,T + r)kwkβ,TTβ−α. Then
kJ.,α,r(F, w)kα,T 6 ρw(T, r) and
Nα,T(J.,α,r(F, w)) 6 (1 + Tα)ρw(T, r).
Proof. Since α < β, the first inequality is an immediate consequence of Propo-sition 3.17 and the obvious inequality kF kα,T 6 Tβ−αkF k
β,T for any F ∈ Cβ-H¨ol([0, T ];Pck(Re)). The second inequality follows, using thatR00f (s) dw(s) = 0 for any f ∈ Sα,r(F ).
4. Existence of fixed-points for functionals of Aumann-Young’s inte-gral and applications to differential inclusions
The main theorem of this section, derived from Kakutani-Fan-Glicksberg’s theorem thanks to the results of Section 3, deals with the existence of fixed-points for functionals of the Aumann-Young integral. We provide some appli-cations to differential inclusions, especially differential inclusions driven by a fractional Brownian motion.
In the sequel, α, β ∈]0, 1[, α + β > 1 and α < β. This implies that β > 12. As usual, T > 0 and w ∈ Cβ-H¨ol([0, T ]; Rd).
4.1. Fixed point theorem
Let us first recall the notions of fixed-points and of upper semicontinuity for multifunctions.
Definition 4.1. Consider a set S and a multifunction F : S ⇒ S. An element x of S is a fixed-point of F if x ∈ F (x).
We now give a definition of upper semicontinuity for multifunctions in a particular case which is sufficient for our needs (see [3, Definition 1.4.1 and Proposition 1.4.8]).
Definition 4.2. Let X be a metric space, let Y be a compact metric space. Let F : X ⇒ Y be a multifunction with closed values, and let S be a closed subset of X, with S ⊂ Dom F := {x ∈ X ; F (x) 6= ∅}. The multifunction F is said to be upper semicontinuous on S if, for every sequence (xn, yn)n∈N of elements of S × Y and for every (x, y) ∈ S × Y such that (xn, yn) converges to (x, y) and yn∈ F (xn) for every n, we have y ∈ F (x).
Let us now set the scene for the fixed point theorem. Let S be a convex compact subset of Cα-H¨ol
([0, T ]; Re
), and let Φ : [0, T ] × Re→ Pck(M`,d
(R))) be continuous for the Hausdorff distance, with ` ∈ N∗. Assume that there exists R > 0 such that
Φ(., x(.)) ∈ Bα,Pck(M`,d(R))(0, R) ; ∀x ∈ S.
Let
r > sup x∈S
minke`kΦ(., x(.))kα,T, kΦ(., x(.))kα,T ,Dem, so that the map
Φw: S −→ Cα-H¨ol([0, T ]; P ck(R`)) x 7−→ (Aα,r) Z . 0 Φ(s, x(s)) dw(s)
is well-defined. With the notations of Corollary 3.18, we have kΦw(x)kα,T 6 ρw(T, r)
for all x ∈ S.
Theorem 4.3 (Fixed point theorem). Let S, Φ, r and Φw as above, let α0 ∈ ]0, α[, and let
Ψ : Cα-H¨ol([0, T ]; Pck(R`)) ⇒ Cα-H¨ol([0, T ]; Re),
a multifunction such that Dom Ψ contains B := {Φw(x) ; x ∈ S}, and which satisfies the following conditions:
1. For every sequence (Fn)n∈N of elements of B such that there exists F ∈ B satisfying Ls n→∞Fn(t) ⊂ F (t) ; ∀t ∈ [0, T ], if ψn ∈ Ψ(Fn) converges in Cα0-H¨ol ([0, T ]; Re) to ψ ∈ Cα-H¨ol([0, T ]; Re), then ψ ∈ Ψ(F ).
2. For every F ∈ B, Ψ(F ) is convex, closed and contained in S.
Then, Γ = Ψ ◦ Φw has at least one fixed-point in S.
Remark 4.4. By Corollary 3.18, in order that Dom(Ψ) ⊃ B, it is sufficient that Dom(Ψ) ⊃ Bα,Pck(M`,d(R))(0, (1 + T
α)ρ
w(T, r)).
Proof of Theorem 4.3. By Condition 2, Γ(x) is closed convex and contained in S for every x ∈ S.
Let us check that Γ is upper semicontinuous. Let (xn)n∈N be a sequence of elements of S converging to x ∈ S in X := Cα0-H¨ol
([0, T ]; Re), and consider ψn∈ Γ(xn) = Ψ(Φw(xn)) converging to ψ ∈ Cα-H¨ol([0, T ]; Re) in X. By Proposition 3.13 (more precisely Remark 3.16) and the Hausdorff continuity of Φ (which gives Ls n→∞Φ(t, xn(t)) ⊂ Φ((t, x(t)) ; ∀t ∈ [0, T ])), we have Ls n→∞Φw(xn)(t) ⊂ Φw(x)(t) ; ∀t ∈ [0, T ].
Then, by Condition 1, ψ ∈ Ψ(Φw(x)) = Γ(x), which proves the upper semicon-tinuity.
We deduce by Kakutani-Fan-Glicksberg Theorem [13, Theorem 8.6 of II.§7] that Γ has at least one fixed-point in S.
Remark 4.5. Consider γ ∈]0, 1∧(β/α)] αγ +β > 1. The statement of Theorem 4.3 remains true when
Φ(., x(.)) ∈ Bαγ,Pck(M`,d(R))(0, R) ; ∀x ∈ S and Φw(x) := (Aαγ,r) Z . 0 Φ(s, x(s)) dw(s).
For the sake of readability, this result has been detailed in the case γ = 1, but the proof of Theorem 4.3 remains unchanged for γ 6= 1, again with S ⊂ Cα-H¨ol
([0, T ]; Re), but replacing α by αγ everywhere else.
Let us provide two examples of multifunctions Ψ fulfilling Conditions 1 and 2 of Theorem 4.3. These examples will be the basis for our applications to differential inclusions.
Example 4.6. Assume that T satisfies ρw(T, r) 6 r with r > 0. With the notations of Theorem 4.3, let us show that Ψh(.) := h + {x ∈ Sα,ρw(T ,r)(·) :
x(0) = 0}, with ` = e and h ∈ Cα-H¨ol
([0, T ]; Re), fulfills Conditions 1 and 2 for S = Sh,r := {x ∈ Bα,Re(h, r) : x(0) = h(0)} which is a convex compact subset
of X := Cα0-H¨ol
([0, T ]; Re) thanks to [12, Proposition 5.28]: 1. Let (Fn)n∈N be a sequence in Bα,Pck(Me,d(R))(0, (1 + T
α)ρw(T, r)) such that there exists F ∈ Bα,Pck(Me,d(R))(0, (1 + T
α)ρ
w(T, r)) satisfying Ls
n→∞Fn(t) ⊂ F (t) ; ∀t ∈ [0, T ]. (12) Consider also ψn ∈ Ψh(Fn) converging in X to ψ ∈ Cα-H¨ol([0, T ]; Re). For any n ∈ N, (ψn− h)(0) = 0 and, by the definition of Sα,ρw(T ,r)(Fn), ψn− h ∈ S
0(F n) and Nα,T(ψn− h) 6 ρw(T, r). Thanks to (12), ψ − h ∈ S0(F ), and since ψ is the limit of ψn in X, (ψ − h)(0) = 0 and Nα,T(ψ − h) 6 ρw(T, r). Therefore, ψ ∈ Ψh(F ).
2. For any F ∈ Bα,Pck(Me,d(R))(0, (1 + T
α)ρw(T, r)), since S
α,ρw(T ,r)(F ) is
convex (resp. closed) by Proposition 3.4 (resp. Proposition 3.6), Ψh(F ) is convex (resp. closed). Moreover, since ρw(T, r) 6 r,
Ψh(F ) ⊂ h + S0(F ) ∩ {x ∈ Bα,Re(0, (1 + Tα)ρw(T, r)) : x(0) = 0}
⊂ h + {x ∈ Bα,Re(0, r) : x(0) = 0} = Sh,r.
Example 4.7. Assume that d = ` = 2, e = 1 and that T satisfies ρw(T, r) 6 r with r > 0. Consider f ∈ Cα-H¨ol([0, T ]; R). With the notations of Theorem 4.3 and Example 4.6, let us show that Ψh,f(.) := h + {f x1− x2 ; x = (x1, x2) ∈ Ψ0(.)} fulfills Conditions 1 and 2 of Theorem 4.3 for S = Sh,f,r := h + {f x1− x2 ; x = (x1, x2) ∈ S0,r}: 1. Let (Fn)n∈Nbe a sequence in Bα,Pck(M1,2(R))(0, (1 + T α)ρ w(T, r)), and let F ∈ Bα,Pck(M1,2(R))(0, (1 + T α)ρw(T, r)) satisfying Ls n→∞Fn(t) ⊂ F (t) ; ∀t ∈ [0, T ]. (13) Consider also ψn ∈ Ψh,f(Fn) converging in X to ψ ∈ Cα-H¨ol([0, T ]; R). For any n ∈ N, ψn = h+f x1,n−x2,nwith xn = (x1,n, x2,n) ∈ S0(Fn) such that xn(0) = 0 and Nα,T(xn) 6 ρw(T, r). Then, by Friz and Victoir [12, Proposition 5.28], there exists a subsequence (xnk)k∈N of (xn)n∈N converging in X to x = (x1, x2) ∈
Cα-H¨ol
([0, T ]; R) such that x(0) = 0 and Nα,T(x) 6 ρw(T, r). Moreover, thanks to (13), x ∈ S0(F ). So, for every t ∈ [0, T ],
ψ(t) = lim n→∞ψn(t) = limk→∞ψnk(t) = h(t) + f (t) lim k→∞x1,nk(t) − limk→∞x2,nk(t) = h(t) + f (t)x1(t) − x2(t). Therefore, ψ ∈ Ψh,f(F ). 2. For any F ∈ Bα,Pck(M1,2(R))(0, (1 + T α)ρw(T, r)), since Ψ0(F ) is convex (resp. Ψ0(F ) ⊂ S0,r), Ψh,f(F ) is convex (resp. Ψh,f(F ) ⊂ Sh,f,r). Moreover, the same arguments than in the previous step yield that Ψh,f(F ) is closed.
4.2. Applications to differential inclusions
Let us provide two applications of Theorem 4.3 to differential inclusions. First, let Φ : [0, T ] × Re → Pck(Me,d
(R)) be a multifunction such that, for every t ∈ [0, T ] and every x ∈ Re, Φ(., x) is α-H¨older continuous with respect to the Hausdorff distance and Φ(t, .) is Lipschitz continuous with respect to the Hausdorff distance too, that is, there exist k1, k2 > 0 such that for every s, t ∈ [0, T ] and x, y ∈ Re,
dH(Φ(s, x), Φ(t, x)) 6 k1|t − s|α and dH
(Φ(t, x), Φ(t, y)) 6 k2kx − yk. (14) Assume also that Φ is bounded with respect to the Hausdorff distance, that is, there exists R > 0 such that
sup (t,x)∈[0,T ]×Re
sup y∈Φ(t,x)
kyk 6 R. (15)
Consider w ∈ Cβ-H¨ol([0, T ]; Rd) and an inclusion of the form
x(t) ∈ ξ + (Aα,r) Z t
0
Φ(s, x(s)) dw(s) ; t ∈ [0, T ], (16)
where r is large enough and the unknown function x is in Cα-H¨ol([0, T ]; Re). Corollary 4.8 (First order differential inclusion). Assume that 0 < α < β, α + β > 1 and r > r0 := R + k1+ k2. Then, the set of solutions to (16) is nonempty.
Proof. First, with the notations of Example 4.6, for any x ∈ Sξ,1, the map s 7→ Φ(s, x(s)) is α-H¨older continuous. Precisely, for every s, t ∈ [0, T ],
dH(Φ(t, x(t)), Φ(s, x(s))) 6 dH(Φ(t, x(t)), Φ(s, x(t))) + dH(Φ(s, x(t)), Φ(s, x(s))) 6 k1|t − s|α+ k2kx(t) − x(s)k 6 (k1+ k2kx − ξkα,T)|t − s|α 6 (k1+ k2)|t − s|α and then, Nα,T(Φ(., x(.))) 6 R + k1+ k2= r0.
Let 1 > T0> 0 be such that (1 + T0α)ρw(T0, r0) 6 1 (see Remark 4.4). Applying Theorem 4.3 on [0, T0], with r = r0, S = Sξ,1 and Ψ = Ψξ (see Example 4.6), shows that Γ = Ψ ◦ Φwhas a fixed point on [0, T0]. Since the definition of T0is independent of ξ, gluing solutions on successive intervals provides a fixed point for Γ on [0, T ], which is thus a solution to (16) on [0, T ].
Remark 4.9. Consider γ ∈]0, 1 ∧ (β/α)] such that αγ + β > 1. Thanks to Remark 4.5, the statement of Corollary 4.3 remains true when, for every t ∈ [0, T ], Φ(t, .) is γ-H¨older continuous but not necessarily Lipschitz continuous.
Now, for e = 1, consider w0∈ Cβ-H¨ol
([0, T ]; R) and a second order inclusion of the form x(t) ∈ ξ + (Aα,ρw0(T ,r)) Z t 0 (Aα,r) Z s 0
Φ(u, x(u)) dw0(u)
dw0(s). (17)
For any x ∈ Cα-H¨ol
([0, T ]; R) such that the Aumann-Young integral in Inclusion (17) is well defined, thanks to the integration by parts formula for Young’s integral, (Aα,ρw0(T ,r)) Z t 0 (Aα,r) Z s 0
Φ(u, x(u)) dw0(u) dw0(s) = Z t 0 Z s 0 ϕ(u) dw0(u) dw0(s) ; ϕ ∈ Sα,r(Φ(., x(.))) = w0(t) Z t 0 ϕ(s) dw0(s) − Z t 0 w0(s)ϕ(s) dw0(s) ; ϕ ∈ Sα,r(Φ(., x(.))) = ( w0(t) Z t 0 ϕ(s)(dw0(s), w0(s) dw0(s)) 1 − Z t 0 ϕ(s)(dw0(s), w0(s) dw0(s)) 2 ; ϕ ∈ Sα,r(Φ(., x(.))) ) .
Then, proving the existence of solutions to (17) amounts to prove that
Γ : x 7−→ Ψh,w0 (Aα,r) Z . 0 Φ(s, x(s)) dw(s) with w := w0, Z . 0 w0(s) dw0(s)
has fixed points.
Corollary 4.10 (Second order differential inclusion). Assume that α < β, α + β > 1 and r > rw0,T := k1+ k2(Nα,T(w0) + 1). Then, the set of solutions
to (17) is nonempty.
Proof. First, consider xw0,ξ:= ξ + w0x1− x2with x = (x1, x2) ∈ S0,1. The map s 7→ Φ(s, xw0,ξ(s)) is α-H¨older continuous. Precisely, for every s, t ∈ [0, T ],
dD(Φ(t, xw0,ξ(t)), Φ(s, xw0,ξ(s))) 6 dD(Φ(t, xw0,ξ(t)), Φ(s, xw0,ξ(t))) +dD(Φ(s, xw0,ξ(t)), Φ(s, xw0,ξ(s))) 6 k1|t − s|α+ k2|xw 0,ξ(t) − xw0,ξ(s)| 6 k1|t − s|α+ k 2(|w0(t)(x1(t) − x1(s))| +|(w0(t) − w0(s))x1(s)| + |x2(t) − x2(s)|) 6 (k1+ k2(Nα,T(w0) + 1))|t − s|α and then, Nα,T(Φ(., x(.))) 6 k1+ k2(Nα,T(w0) + 1) = rw ,T.
Let T0> 0 be such that (1 + T0α)ρw(T0, rw0,T) 6 1. By Theorem 4.3 applied on [0, T0], with r = rw0,T, S = Sξ,w0,1and Ψ = Ψξ,w0(see Example 4.7), Γ = Ψ◦Φw
has a fixed point, which is thus a solution to (17) on [0, T0]. Since the definition of T0 is independent of ξ, Γ has a fixed point, which is thus a solution to (17) on [0, T ].
Let us conclude with applications to stochastic inclusions. Consider a (d−1)-dimensional fractional Brownian motion B = (B(t))t∈[0,T ]of Hurst index H ∈ (1/2, 1), which is a centered Gaussian process such that
E(Bi(s)Bj(t)) = 1 2(t
2H+ s2H− |t − s|2H)δ i,j
for every s, t ∈ [0, T ] and i, j ∈ {1, . . . , d − 1}, and let (Ω, A, P) be the associ-ated canonical probability space. By the Garcia-Rodemich-Rumsey lemma (see Nualart [28, Lemma A.3.1]), the paths of B are β-H¨older continuous for any β ∈ (0, H). Consider r > 0 and α ∈ (0, 1) such that α + β > 1 and α < β. Then, for any measurable map F : Ω → Bα,Pck(Me,d(R))(0, r), one can define a
set-valued stochastic integral of F with respect to B by (Aα,r) Z t 0 F (s)dB(s) (ω) := (Aα,r) Z t 0 F (s, ω)dB(s, ω) ; ω ∈ Ω, t ∈ [0, T ].
This allows to consider the stochastic inclusion
X(t) ∈ ξ + (Aα,r) Z t
0
Φ(s, X(s)) dw(s) ; t ∈ [0, T ], (18)
where Φ : [0, T ] × Re → Pck(Me,d
(R)) fulfills Assumptions (14) and (15), and W (t) := (t, B1(t), . . . , Bd−1(t)) for every t ∈ [0, T ]. By Corollary 4.8, for every r > k1+ k2+ R, Inclusion (18) has at least one pathwise solution. One can also consider the one-dimensional second order stochastic inclusion
X(t) ∈ ξ + (Aα,ρB(T ,r)) Z t 0 (Aα,r) Z s 0 Φ(u, X(u))dB(u) dB(s) ; t ∈ [0, T ]. (19) By Corollary 4.8, for every r > k1+ k2(Nα,T(B) + 1), Inclusion (19) has at least one pathwise solution.
Acknowledgments. This work was funded by RFBR and CNRS, project number PRC2767. We also thank the GDR TRAG (CNRS) for its support.
References
[1] J. ao Guerra and D. Nualart. Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stochastic Anal. Appl., 26(5):1053–1075, 2008.
[2] J.-P. Aubin and A. Cellina. Set-Valued Maps and Viability Theory. Springer, 1984.
[3] J.-P. Aubin and H. Frankowska. Set-valued analysis, volume 2. Boston etc.: Birkh¨auser, 1990.
[4] R. J. Aumann. Integrals of set-valued functions. J. Math. Anal. Appl., 12:1–12, 1965.
[5] R. Baier and E. Farkhi. Regularity and integration of set-valued maps represented by generalized Steiner points. Set-Valued Anal., 15(2):185–207, 2007.
[6] I. Bailleul, A. Brault, and L. Coutin. Young and rough differential inclu-sions. ArXiv e-prints, 2020.
[7] G. Beer. Topologies on closed and closed convex sets. Dordrecht: Kluwer Academic Publishers, 1993.
[8] J. P. R. Christensen. Topology and Borel structure. North-Holland Publish-ing Co., Amsterdam-London; American Elsevier PublishPublish-ing Co., Inc., New York, 1974. Descriptive topology and set theory with applications to func-tional analysis and measure theory, North-Holland Mathematics Studies, Vol. 10. (Notas de Matem´atica, No. 51).
[9] S. Cox, M. Hutzenthaler, A. Jentzen, J. van Neerven, and T. Welti. Con-vergence in H¨older norms with applications to Monte Carlo methods in infinite dimensions. IMA Journal of Numerical Analysis, 41(1):493–548, 2020.
[10] G. Debreu. Integration of correspondences. Proc. 5th Berkeley Symp. Math. Stat. Probab., Univ. Calif. 1965/66, 2, Part 1 351-372 (1967)., 1967. [11] D. Dentcheva. Regular Castaing representations of multifunctions with applications to stochastic programming. SIAM J. Optim., 10(3):732–749, 2000.
[12] P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths. Theory and applications, volume 120. Cambridge: Cambridge University Press, 2010.
[13] A. Granas and J. Dugundji. Fixed point theory. New York, NY: Springer, 2003.
[14] J.-B. Hiriart-Urruty and C. Lemar´echal. Convex analysis and minimization algorithms. Part 1: Fundamentals, volume 305. Berlin: Springer-Verlag, 1993.
[15] M. Hukuhara. Int´egration des applications mesurables dont la valeur est un compact convexe. Funkc. Ekvacioj, Ser. Int., 10:205–223, 1967. [16] E. J. Jung and J. H. Kim. On set-valued stochastic integrals. Stochastic
[17] M. Kisielewicz. Set-Valued Stochastic Integrals and Applications. Springer, 2020.
[18] M. Kisielewicz. Set-valued stochastic integrals and stochastic inclusions. Stochastic Anal. Appl., 15(5):783–800, 1997.
[19] M. Kisielewicz. Some properties of set-valued stochastic integrals. J. Math. Anal. Appl., 388(2):984–995, 2012.
[20] M. Kisielewicz and J. Motyl. Selection theorems for set-valued stochastic integrals. Stochastic Anal. Appl., 37(2):243–270, 2019.
[21] A. A. Levakov and M. M. Vas’kovskii. Existence of solutions of stochas-tic differential inclusions with standard and fractional Brownian motions. Differ. Equ., 51(8):991–997, 2015.
[22] M. T. Malinowski and M. Michta. Set-valued stochastic integral equations. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, 18(4):473– 492, 2011.
[23] M. T. Malinowski, M. Michta, and J. Sobolewska. Set-valued and fuzzy stochastic differential equations driven by semimartingales. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 79:204–220, 2013. [24] M. Michta. Remarks on unboundedness of set-valued Itˆo stochastic
inte-grals. J. Math. Anal. Appl., 424(1):651–663, 2015.
[25] M. Michta. Stochastic integrals and stochastic equations in set-valued and fuzzy-valued frameworks. Stoch. Dyn., 20(1):47, 2020. Id/No 2050001. [26] M. Michta and J. Motyl. Selection Properties and Set-Valued Young
Inte-grals of Set-Valued Functions. Results Math, 75:1–22, 2020.
[27] M. Michta and J. Motyl. Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals. J. Theor. Probab., page 23 pages, 2020.
[28] D. Nualart. The Malliavin Calculus and Related Topics. Springer, 2006. [29] T. Rze˙zuchowski. The Demyanov metric and some other metrics in the
family of convex sets. Cent. Eur. J. Math., 10(6):2229–2239, 2012. [30] J. Saint Pierre. Point de Steiner et sections lipschitziennes. (Steiner point
and Lipschitz sections). S´emin. Anal. Convexe, Univ. Sci. Tech. Languedoc, 15:42, 1985.
[31] R. A. Vitale. The Steiner point in infinite dimensions. Isr. J. Math., 52:245–250, 1985.
